Understand and Represent Linear Functions

A.PA.07.06 Calculate the slope from the graph of a linear function as the ratio of "rise/run" for a pair of points on the graph, and express the answer as a fraction and a decimal; understand that linear functions have slope that is a constant rate of change. A.PA.07.07 Represent linear functions in the form y = x + b, y = mx, and y = mx + b, and graph, interpreting slope and y-intercept. A.FO.07.08 Find and interpret the x and/or y intercepts of a linear equation or function. Know that the solution to a linear equation of the form ax+b=0 corresponds to the point at which the graph of y=ax+b crosses the x axis.

Unwrapping the GLCE’s:Concepts (what students need to know):

Rise/run Slope (positive, negative, undefined, zero) Constant rate of change Linear function X & Y-intercept Slope intercept form equation

Skills (what students need to be able to do): Determine and interpret X & Y-intercepts Given a table for a linear function, a line on a graph, or the coordinates of

two points, find the slope and linear equation. Recognize and indentify the four types of slope.

Prior Knowledge: Change fractions to decimals, decimals to fractions Graph points on coordinate grid Understand algebraic equations & solve Transfer information from tables to graphs, graphs to tables Finding rates of change from word problem and graph

Big Ideas:1. Finding rise/run can be done mathematically by finding the difference in y values and

x values. 2. X and Y intercepts can be found both by reading the graph and by using the equation.3. Slope and linear equations can be used to solve real world problems.

Essential Questions:1. What is slope?2. What do each of the types of slope look like when graphed?3. How can slope be determined without counting on a graph?

4. How can X and Y intercepts be determined without a graph?5. How are slope and linear equations used to solve real world problems?

Post Assessment Questions (without answers):Graph a line with a slope of -1/4.

Graph a line with a slope of 0.

Graph a line with an undefined slope.

Graph a line with a slope of 5/4.

Find the error:Mike and Chloe are finding the slope of the line that passes through Q(-2,8) and I(11,7).

Mike Chloe m= 8-7 m= 7-8

-2-11 11-2

Who is correct? Explain your reasoning.

Find the x-intercept and y-intercept of each equation:y= x+ 8 y= -3x+6 4x-y = 12

Fingernails grow at a constant rate. In the linear equation y= 1.5 x + .5, y represents the total length of your nails(inches) after x number of years.

a. What is the meaning of the slope in the equation?

b. What is the meaning of the y-intercept in the equation?

Pre-Assessment Questions(without answers):

Circle the correct graph above.

This is a two-part question. Draw a graph through the points (3,4) and (7,12).

Second, select the correct slope of this line.

A.

B.

C.

D.

Which of the following is the best graph of y = -2x ?

Circle the correct graph below.

 

A.

Given the points (3, 8) and (8, 12) calculate the slope of the line that the points lie on.

 

A.

B.

C.

D.

Post Assessment Questions (with answers):Graph a line with a slope of -1/4 (4 points).

An Example: Slope passes between the points A (0,0) and B(4,-1).

Graph a line with a slope of 0.

An Example: A line passes through point A(0,6).

Graph a line with an undefined slope.

An Example: A line passes through point B(-4,0).

Graph a line with a slope of 5/4.

An Example: A line passes through points C(0,0) and D(4,5).

Find the error:Mike and Chloe are finding the slope of the line that passes through Q(-2,8) and I(11,7).

Mike Chloe m= 8-7 m= 7-8 -2-11 11-2

Who is correct? Explain your reasoning.

A (0,0)

B (4,-1)

A(0,6)

ANSWER: Mike is correct. Chloe did not subtract negative two. She subtracted two. To find the slope of a line, find the change in Y divided by change in X.

Find the x-intercept and y-intercept of each equation:y= x+ 8 y= -3x+6 4x-y = 12x= -8 x= -2 x=+3y=+8 y=+6 y=-8

Fingernails grow at a constant rate. In the linear equation y= 1.5 x + .5, y represents the total length of your nails after x number of years.

c. What is the meaning of the slope in the equation?

ANSWER: A person’s nails grow 1.5 inches per year.

d. What is the meaning of the y-intercept in the equation?

ANSWER: Y=0.5 This represents the starting length of a person’s nails.

Pre-Assessment Questions (with answers):

ANSWER: B

ANSWER: A

Circle the correct graph above.ANSWER: Graph B

ANSWER: D

1. This is a two-part question. Draw a graph through the points (3,4)

and (7,12).

ANSWER: plot (3,4) and (7,12) draw a line…Second, select the correct slope of this line.

E.

F.

G.

H.

ANSWER: C

3. Which of the following is the best graph of y = -2x ?

 

A. A B. B

C. C

D. D

ANSWER: A

B.

2. Given the points (3, 8) and (8, 12) calculate the slope of the line that the points lie on.

 

E.

F.

G.

H.

ANSWER: C

INSTRUCTIONAL LESSON AND ACTIVITIES:

Day One:

Use the following activity to introduce your unit:

Introductory activity (http://answers.yahoo.com/question/index?qid=1006011403697):

This year I used an activity suggested in an article in Mathematics Teaching in the Middle School that used a "book-stacking" model to present the concepts of slope and intercept.

A group of students started stacking their textbooks on a surface of the desk and measuring the total height of the stack. Relating the height of the books to the number of books in the stack provided a linear graph with the slope being represented by the height of one book (when we add one book, how does the height change?), and the intercept being represented by the height of the desk (when there are no books, what is the height? If we stacked books on top of the filing cabinet instead, how would the height change?)

The students were led to explore the slope-intercept form of a linear equation through a predictable physical situation before these concepts were taught. I had reasonably good results with a group of lower-ability and lower-motivated students.

Also, to introduce unit, discuss roof slopes, stairs, and road signs indicating slope of incline. Also review vertical, horizontal, and coordinate graphing skills.

Remind students how their found rates of change, both positive and negative.

Define slope as rise/run, or the ratio of the rise, or vertical change, to the run, or horizontal change. Using a graph paper on white board, plot two points from their coordinate pairs. Connect the two points with a line. Demonstrate how to count the rise and run between the two points. Show them how it is either a positive or negative slope.

Give students either graph paper or individual whiteboards with grids on it, and have them plot points as large group activity, finding slopes for each set of points.

Give students independent practice on this skill using premade graphs.

Day Two:

Go over independent work for review of finding slope from graph.

Looking at the independent work, ask students if they can come up with a way to find slopes without actually counting. Coach them to see the rise is the difference in y-values, and the run is the difference in x-values. This should be an easy connection from rates of change. Then, introduce using values in a table to find slope.

Introduce slope formula: m= y2-y1

x2-x1

Give students several tables of values to practice from as large group.

Give students independent practice on this skill using premade tables.

Day Three:

Review finding slope from a table by going over independent work from yesterday.

Use one of the online activities for students to practice:

Slope Slider http://www.shodor.org/interactivate/activities/SlopeSlider/?version=1.6.0&browser=MSIE&vendor=Sun_Microsystems_Inc.Algebra vs. Cockroachhttp://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

Exploring Slopehttp://www.northcanton.sparcc.org/~technology/excel/files/Exploring_Slope.xls

Play Slope Mania for review:Students should be divided into groups of 2-3. Each group will need a large laminated

sheet of grid paper, a dry erase marker, a meter stick, and 4 dice (2 each of 2 different colors, one of each color should be marked with a negative sign). They will also need the handout (at end of unit plan) for recording their rolls and finding their slopes.

Day Three/Four/Five:

Use one or more of the following Illuminations activities to apply slope to a real world situation:

Barbie Bungeehttp://illuminations.nctm.org/LessonDetail.aspx?id=L646Constant Dimensionshttp://illuminations.nctm.org/LessonDetail.aspx?id=L572Printing Bookshttp://illuminations.nctm.org/LessonDetail.aspx?id=L271Walk the Plankhttp://illuminations.nctm.org/LessonDetail.aspx?id=L682

Day Six:

Use MiGlance Lessons – 7th grade – Session 5 – Walkathon Pledge Plan

Instructional Activities: Use math text to teach rates of change, rise/run using both a graph and coordinates of two points, slope, and linear equations/functions.For example:~Using Glencoe Pre-Algebra, see Chapter 8~Using Glencoe Mathematics: Applications and Concepts, Course 3, see Chapter 4~ Using Prentiss Hall Mathematics Course 2, see Chapter 10~Using UCSMP Algebra, Chapter 7

United Streaming videos:Operation Algebra: Slope of a LineLinear Equations: y = mx + b

NAMES_______________________________________________________________________

Slope Mania

Together with your partner(s), roll your 4 dice. Separate into 2 pairs, each with one red and one green die. Record the values below. On your graph poster, place a bingo chip at each point. Record your roll as well as your ordered pairs on the table. Calculate your slope by counting your RISE/RUN. Record on the table. Use the meter stick to help you estimate the Y-intercept. If it is not EXACT, tell which 2 numbers it

falls between. Repeat until you have calculated 10 different slopes and Y-intercepts.

Red Die Value #1(X)

Green Die Value #1(Y)

Ordered Pair of Point #1(X,Y)

Red Die Value #2(X)

Green Die Value #2( Y)

Ordered Pair of Point #2(X,Y)

Y2 – Y1

X1-X2

Slope (m)

Y-Intercept (b)